The suspect wanted in the slaughter of his Brooklyn girlfriend and her beautiful daughter is a Russian army vet trained to strategically kill and vanish — and he fled to his homeland because he knew its laws would protect him from prosecution, sources said yesterday.

Russia and the United States have no extradition treaty, but even if they did, Russian law prevents its citizens from being sent abroad for prosecution.

That means special-ops-soldier-turned-day-laborer Nikolai Rakossi will likely get away with the vicious killing of his girlfriend, Tatyana Prikhodko, 56, and her stunning daughter Larisa, 28.

“According to the Russian constitution, Russian citizens can’t be extradited to [a] foreign country,” Yevgeniy Khorishko, spokesman for the Russian Embassy in Washington, told The Post.

Despite the red tape, outraged authorities in New York are working with the State Department to see if they can convince Russia to say da instead of nyet once an arrest warrant is issued.

“It’s god-awful!” fumed an NYPD detective. “I wish that our government would have more cooperation with the Russians. If he committed the crime here, he should pay here.”

Rakossi, 52, boarded a 7 p.m. Aeroflot flight to Moscow on Sunday, less than five hours before cops found registered nurses Tatyana and Larisa brutally stabbed to death in a blood-spattered Sheepshead Bay apartment that he and the older woman shared.

“He’s crazy — he went crazy,” said Larisa’s close friend Yelena Raykin.

On Saturday, a neighbor heard arguing in the East 13th Street apartment — three floors above the unit where Larisa lived with her 2½-year-old son, Ryan, who was staying with his dad, Felix Zeltser.

Larisa failed to show for her night shift at Mount Sinai Hospital, and Rakossi likely spent the night with the two bodies before leaving at around 9:30 a.m. Sunday with two suitcases and catching a cab to JFK, sources said.

“You’ve got to be sick to stay in the same apartment with two dead people,” Raykin said.

Rakossi may be sick, but he is also cunning.

In Russia, he is a well-respected former “elite” member of a special-operations unit. He carried out missions in Afghanistan, Africa and Vietnam, said Rustam Zaripon, a manager of the Russian Baths in Sheepshead Bay.

“He has a home outside of Moscow, and he’s a champion out there,” said Michael Grinberg, 66, another manager at the spa, where Rakossi and Tatyana were regulars until last month.

“He’s a big, strong man and a great wrestler,” Grinberg said. “He came in every Sunday and would give candy to the kids and be friendly with everyone. I thought I knew him as a friend, but now I don’t think I knew him as well as I [thought I] did.”

A former federal prosecutor said that it is “a long shot” that the White House or State Department will ask Russia for a favor by returning Rakossi in the absence of an extradition treaty.

But he noted that the Justice Department could prepare an arrest warrant for unlawful flight to avoid prosecution, to detain him if he travels to a country that does have an extradition agreement with the United States.

Meanwhile, Larisa’s sister Svetlana Shrifrina arrived from Switzerland to identify the two women’s bodies and to make funeral arrangements.